UNITED NATIONS (CNN) -- Russian forces will be moving out of Georgia by Friday night, Russia's ambassador to the United Nations said Thursday.

"The idea is, by tomorrow night, all those forces ... will move to within that perimeter, the peacekeeping perimeter," Vitaly Churkin told reporters.

That assessment jibed with one offered earlier Thursday in Moscow, where the deputy chief of staff of Russia's armed forces laid out a similar timeline.

"By the end of the 22nd, we will pull back to the checkpoints line," Gen. Anatoly Nogovitsyn told a military briefing in Moscow, according to Russia Today Television.

But the commander of Russia's land forces, Gen. Vladimir Boldyrev, told Russia's Interfax news agency later Thursday that his troops would be back on Russian soil in 10 days.

Boldyrev said Russian peacekeeping troops would be stationed at posts that troops have been constructing -- some of them inside Georgian territory -- since Russian troops crossed the border on August 8. Russia argues that it is allowed to expand its security zone under a 1992 agreement.

Churkin offered yet another scenario for yet another timetable.

"Some military commanders think it might take 10 days for that withdrawal to take place," he said outside the U.N. Security Council, where representatives worked Thursday on different versions of a resolution intended to resolve the crisis.

Russia's incursion into the former Soviet republic followed the launch of a Georgian campaign against the Russian-backed separatist territory of South Ossetia on August 7. Russian tanks, troops and armored vehicles poured into South Ossetia and another breakaway Georgian territory, Abkhazia, the next day, advancing into Georgian cities across the administrative borders within those regions.

The two sides have constantly blamed each other for starting the conflict as well as for a wide variety of offenses leading up to and during the fighting, including ethnic cleansing.

Russia and Georgia signed a French-brokered, six-point cease-fire agreement last weekend that allows Russian forces to establish a buffer zone inside Georgia within a few kilometers of South Ossetia. Under international agreements, Russian peacekeepers have been stationed in the pro-Moscow breakaway republic for more than a decade.

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The Russian version of a U.N. Security Council resolution on the matter, Churkin said, "simply repeats verbatim the document which was adopted by [Russian President] Dmitry Medvedev and [French President Nicolas] Sarkozy."

Asked why Russia's draft contains no reference to the issue of territorial integrity, Churkin reiterated that it was simply a repetition of the six principles agreed to by the presidents of Russia and France. "And you know what?" he asked rhetorically. "There's no territorial integrity in the six principles."

Asked about reports that Russian forces are in the port city of Poti, deep within Georgian territory, he said, "I have no information on Poti. I have to double-check."

Permanent U.S. Representative to the United Nations Alejandro Daniel Wolff said the principle of territorial integrity must be dealt with in any resolution that will win his vote.

"The six points alone, without clarifications and without signs of compliance with those six points as clarified raise the question why we put the council in the position of enshrining something that's not being adhered to," he said. "It's a fundamental principle."

Top Russian officials have repeatedly said their troops are pulling back in accordance with the cease-fire.

But U.S. Secretary of State Condoleezza Rice told CNN on Wednesday that Russian forces remain in Georgian cities, and that Medvedev "is beginning to sound like a broken record."

"First his troops are going to be out on Monday. Then his troops were gonna be out on Wednesday. Now his troops are gonna be out on Friday," Rice said. "I'm beginning to wonder if the Russian president is ever going to keep his word, or can he keep his word, or what's going on there."

In a telephone call Thursday to Georgia's President Mikheil Saakashvili, President Bush said he expects Russia to withdraw its forces from the former Soviet republic and end "their siege of Georgia," a White House spokesman said.

Washington has been pushing its allies to isolate Russia diplomatically over the incursion by suspending NATO-Russian contacts. In addition, Rice said, the world's leading industrial powers are not likely to meet with Russia as the Group of Eight in the near future.

Also, the U.S.-based human rights monitoring organization, Human Rights Watch, said in a statement Thursday that Russian attacks used cluster munitions, "refuting Russia's earlier denials that it used the weapon."

"Many people have died because of Russia's use of cluster munitions in Georgia, even as Moscow denied it had used this barbaric weapon," it said.

Meanwhile, investigations of ethnic cleansing charges filed by the Republic of Georgia against the Russian Confederation are under way.

The legal counsel for the Georgian government, the Canadian law professor Payam Akhayan, was to travel Thursday to Georgia to gather evidence for a case he filed August 12 at the International Court of Justice in The Hague.

The International Court has scheduled a hearing for early September.

The Russian invasion of Georgia's breakaway region had been "premeditated and planned in great detail," Akhayan said.

He dismissed assertions by Moscow that Georgia itself was responsible for ethnic cleansing against minority groups in South Ossetia and Abkhazia. "The Georgian offensive was about one day long," he said. "How long has Russian been occupying Georgia? We have to think about that."

Akhayan also accused the Russian government of a 17-year "campaign of terror" against ethnic Georgians. The case includes allegations of human rights abuses dating back to 1993.